\section{Models}
\label{sec:risk.models}

We obtain our results through both analytical techniques and
simulations on a range of networks including preferential attachment
networks~\cite{barabasi+perferattach99} and large synthetic and
real-world networks.  For our analyses, we adopt the SIR model of
epidemics defined on networks.  Let $G=(V,E)$ denote an undirected
social contact graph, where $V$ denotes a set of people (referred to
as nodes henceforth) and $(u,v)\in E$ denotes a contact between nodes
$u$ and $v$ (see Figure \ref{fig:mh}(a) for an example).  If node $u$
becomes infectious, it will infect each of its susceptible neighbors
independently with probability $\pt$ (referred as {\em base
transmissivity}).  Each node in the graph is either vaccinated (e.g.,
nodes $B$ or $F$ in Figure \ref{fig:mh}(b)) or not (e.g., nodes $A$ or
$C$ in Figure \ref{fig:mh}(b)).  If a node $u$ is not vaccinated, we
label it as {\uv}. The vaccine fails with probability $\pf$. If a node
$u$'s vaccine fails, we label it as {\vf}; otherwise, we label it
{\vs}. Both {\uv} and {\vf} nodes are susceptible. We assume that
vaccine failure is a stochastic event and that (vaccinated) nodes do
not know if (their own) vaccination succeeded or not. If a node with
vaccine failure is infected then its risk behavior changes, i.e., it
increases its contacts to some of its' neighbors, resulting in {\em
boosted transmissivity}\/ $\pb$ - in the one-sided model a node
infects all its susceptible neighbors with boosted transmissivity
$\pb$, while in the two-sided model, it only infects those neighbors
with boosted transmissivity $\pb$ that have also had a failed
vaccination. In the rest of the paper, we use $\pv$ to denote the
probability that a node is vaccinated, under a campaign of uniformly
random vaccination.

The disease transmission process is thus defined by the tuple
$\rb{\pt, \pb, \pf, \pv}$ in the following manner: every node is
labeled with {\uv}, {\vs}, {\vf} with probability $1-\pv$,
$\pv(1-\pf)$, and $\pv\pf$, respectively. All nodes labeled {\vs} are
removed from the graph. Each edge $(u,v)$ connecting two surviving
nodes $u$ and $v$, is ``open'' (or retained in the graph, in the
language of percolation, which corresponds to disease transmission on
this edge), or ``closed'' (or removed from the graph), with some
probability depending on the model - (i) in the one-sided model, edge
$(u,v)$ is open with probability $\pt$, if both $u$ and $v$ are
labeled {\uv}, and is open with probability $\pb$ if one of $u$ and
$v$ is labeled {\vf}; (ii) in the two-sided model, edge $(u,v)$ is
open with probability $\pt$, unless both $u$ and $v$ are labeled
{\vf}. Following the well known correspondence between bond
percolation and disease transmission, the connected component
containing a specific node $u$ is the (random) subset of nodes
infected, if the disease starts at $u$. If the components resulting
from one random instance of the above stochastic process are $C_1,
C_2,\dots, C_k$, then $\sum_i \abs{C_i}^2/n$ denotes the expected
outbreak size of the disease starting from a random initial node. In
our analysis, we use this as a measure of {\em epidemic severity}.

\iffalse
First we define our model formally as follows. $G=(V,E)$ denotes an
undirected contact graph, where $V$ denotes a set of people (referred
to as nodes) and $(u,v)\in E$ denotes a contact between nodes $u$ and
$v$. If $u$ is infectious, it will infect each of its susceptible
neighbor independently with probability $\pt$ (referred as base
transmissivity). Each node in the graph is either vaccinated or
not. If a node $u$ is not vaccinated, we label it as {\uv}. The
vaccine fails with probability $\pf$. If a node $u$'s vaccine fails,
we label it as {\vf}; otherwise, we label it {\vs}. Both {\uv} and
{\vf} nodes are susceptible. We assume that vaccinated nodes do not
know if the vaccine succeeded or not, and (if infected) they increase
contacts to some of their neighbors, resulting in boosted
transmissivity $\pb$ - in one-sided model, node $u$ infected all its
susceptible neighbors with boosted transmissivity $\pb$, while in
two-sided model, node $u$ only infects neighbor $v$ with boosted
transmissivity $\pb$ if $v$ is vaccinated but failed. In most of the
paper, we assume nodes are vaccinated randomly, with some probability,
denoted as $\pv$.

Therefore, the above disease transmission process is defined by the
tuple $\rb{\pt, \pb, \pf, \pv}$ in the following manner: every node is
labeled with {\uv}, {\vs}, {\vf} with probability $1-\pv$,
$\pv(1-\pf)$, and $\pv\pf$, respectively. All nodes labeled {\vs} are
removed from the graph. Each edge $(u,v)$ connecting two surviving
nodes $u$ and $v$, is ``open'' (or retained in the graph, in the
language of percolation, which corresponds to disease transmission on
this edge), or ``closed'' (or removed from the graph), with some
probability depending on the model - (i) in the one-sided model, edge
$(u,v)$ is open with probability $\pt$, if both $u$ and $v$ are
labeled {\uv}, and is open with probability $\pb$ if one of $u$ and
$v$ is labeled {\vf}; (ii) in the two-sided model, edge $(u,v)$ is open
with probability $\pt$, unless both $u$ and $v$ are labeled
{\vf}. Following the well known correspondence between bond
percolation and disease transmission, the connected component
containing a specific node $u$ is the (random) subset of nodes
infected, if the disease starts at $u$. If the components resulting
from one random instance of the above stochastic process are $C_1,
C_2,\dots, C_k$, then $\sum_i \abs{C_i}^2/n$ denotes the average
outbreak size of the disease starts at a random initial node. In our
analysis, we use this as a measure of epidemic severity. 
\fi

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=5in]{figures/risk-model.jpg}
\caption{Sidedness of risk behavior change: the one-sided and two-sided models. }
\label{fig:mh}
\end{center}
\end{figure}
